In this paper, we consider finite, simple and connected graphs. The vertex and edge sets of a graph \(G\)  are denoted by \(V(G)\)  and \(E(G)\) , respectively. Two vertices \(x,y\in V(G)\)  are resolved (resp. determined or identified) by a vertex \(v\in V(G)\)  if \(d_G(x,v)\ne d_G(y,v)\)  where \(d_G\)  stands for the classical distance operator in graphs. A set \(S\subset V(G)\)  is called a resolving set for \(G\)  if every two vertices \(u,v\in V(G)\)  are resolved by a vertex of \(S\) . A resolving set of the smallest possible cardinality in \(G\)  is called a metric basis, and the cardinality of a metric basis is the metric dimension of \(G\) , denoted \(\dim (G)\) . These concepts were first formally and independently introduced for graphs in [1]}, [2]} in connection with some problems of uniquely recognizing the vertices of a graph with a purpose of locating intruders in a network. We may remark that in [2]}, resolving sets were named locating sets and the metric dimension was called locating number.
